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Properties

Label 16.2.32262667623...0000.2
Degree $16$
Signature $[2, 7]$
Discriminant $-\,2^{20}\cdot 5^{8}\cdot 31^{12}$
Root discriminant $69.87$
Ramified primes $2, 5, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-300899, 3725040, -17128241, -1936548, 30242442, -32990878, 18177768, -5770220, 955169, -20040, -21338, 4176, -910, 250, -17, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 17*x^14 + 250*x^13 - 910*x^12 + 4176*x^11 - 21338*x^10 - 20040*x^9 + 955169*x^8 - 5770220*x^7 + 18177768*x^6 - 32990878*x^5 + 30242442*x^4 - 1936548*x^3 - 17128241*x^2 + 3725040*x - 300899)
 
gp: K = bnfinit(x^16 - 6*x^15 - 17*x^14 + 250*x^13 - 910*x^12 + 4176*x^11 - 21338*x^10 - 20040*x^9 + 955169*x^8 - 5770220*x^7 + 18177768*x^6 - 32990878*x^5 + 30242442*x^4 - 1936548*x^3 - 17128241*x^2 + 3725040*x - 300899, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} - 17 x^{14} + 250 x^{13} - 910 x^{12} + 4176 x^{11} - 21338 x^{10} - 20040 x^{9} + 955169 x^{8} - 5770220 x^{7} + 18177768 x^{6} - 32990878 x^{5} + 30242442 x^{4} - 1936548 x^{3} - 17128241 x^{2} + 3725040 x - 300899 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-322626676239789982105600000000=-\,2^{20}\cdot 5^{8}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.87$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{170061370676483866159612798971212923971141599320028977459} a^{15} - \frac{1105270618115080198163635536416256564832705972510218218}{15460124606953078741782981724655720361012872665457179769} a^{14} - \frac{56893871692421327670865387231488388044857276489920199737}{170061370676483866159612798971212923971141599320028977459} a^{13} + \frac{1523104060483351591110762315833149464168556314562918218}{170061370676483866159612798971212923971141599320028977459} a^{12} + \frac{279837009346671575487228059554374952036046946866524652}{15460124606953078741782981724655720361012872665457179769} a^{11} + \frac{40384959302256756389458476674265450045347361426528616067}{170061370676483866159612798971212923971141599320028977459} a^{10} - \frac{4040638989541934432272885922738794457764615860901450677}{15460124606953078741782981724655720361012872665457179769} a^{9} + \frac{8892805317514654241116210047242094484355346339349523591}{170061370676483866159612798971212923971141599320028977459} a^{8} + \frac{23194929857957272479680835001666582699438941562397779899}{170061370676483866159612798971212923971141599320028977459} a^{7} - \frac{44229306332386619259977786974094579921696695832040298791}{170061370676483866159612798971212923971141599320028977459} a^{6} + \frac{3265547993878968820775603334262396316582950386674722847}{170061370676483866159612798971212923971141599320028977459} a^{5} + \frac{77880102091894870612526115207023487620848280184737710018}{170061370676483866159612798971212923971141599320028977459} a^{4} + \frac{39869081894468930627348293944531208101056180933478982761}{170061370676483866159612798971212923971141599320028977459} a^{3} - \frac{39091865381463212965925005664321984254725019390849555264}{170061370676483866159612798971212923971141599320028977459} a^{2} + \frac{74357568109712160051603275753865130102709136424243433067}{170061370676483866159612798971212923971141599320028977459} a - \frac{33667825590351992588224501788217149389449906786660177122}{170061370676483866159612798971212923971141599320028977459}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 208866867.023 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 29 conjugacy class representatives for t16n1022
Character table for t16n1022 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.775.1, 8.2.9235210000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed
Arithmetically equvalently siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $16$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.8.8.9$x^{8} + 6 x^{6} + 4 x^{5} + 16$$2$$4$$8$$((C_8 : C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
2.8.12.22$x^{8} + 4 x^{7} + 16 x^{3} + 48$$4$$2$$12$$(((C_4 \times C_2): C_2):C_2):C_2$$[2, 2, 2, 2]^{4}$
5Data not computed
$31$31.8.6.2$x^{8} + 713 x^{4} + 138384$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
31.8.6.3$x^{8} - 31 x^{4} + 11532$$4$$2$$6$$C_8:C_2$$[\ ]_{4}^{4}$